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Optimal replacement times for machining tool during turning titanium
metal matrix composites under variable machining conditions
Shaban, Yasser; Aramesh, Maryam; Yacout, Soumaya; Balazinski, Marek;
Attia, Helmi; Kishawy, Hossam
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2017, Vol. 231(6) 924–932 ÓIMechE 2015 Reprints and permissions:
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Optimal replacement times for
machining tool during turning titanium
metal matrix composites under
variable machining conditions
Yasser Shaban
1, Maryam Aramesh
1, Soumaya Yacout
1, Marek Balazinski
1,
Helmi Attia
2and Hossam Kishawy
3Abstract
Little practical results are known about the cutting tool optimal replacement time, specifically for machining of compo-site materials. Due to the fact that tool failure represents about 20% of machine down-time, and due to the high cost of machining, in particular when the work piece’s material is very expensive, optimization of tool replacement time is thus fundamental. Finding the optimal replacement time has also positive impact on product quality in terms of dimensions and surface finish. In this article, two new contributions to research on tool replacement are introduced. First, tool replacement mathematical models are proposed. These models are used in order to find the optimal time to tool replacement when the tool is used under variable machining conditions, namely, the cutting speed and the feed rate. Proportional hazards models are used to find an optimal replacement function. Second, this model is obtained during turning titanium metal matrix composites. These composites are a new generation of materials which have proven to be viable in various industrial fields such as biomedical and aerospace, and they are very expensive. Experimental data are obtained and used in order to develop and to validate the proportional hazards models, which are then used to find the optimal replacement conditions.
Keywords
Optimal tool replacement, metal matrix composites, cost optimization, availability optimization
Date received: 12 March 2014; accepted: 19 November 2014
Introduction
Titanium metal matrix composites (TiMMCs) inherit outstanding characteristics such as low weight, high mechanical and physical properties, high stiffness, and strength. For example, the density of conventional TiMMCs is 4040 kg/m3, and the stiffness is 200 GPa.1 Although very expensive, metal matrix composites (MMCs) are a new generation of materials which have proven to be viable in various fields such as biomedical and aerospace industrial. Finding the optimal tool replacement time in machining TiMMCs is important in order to decrease the scrapped products and thus the cost of machining, and/or to increase the tool life, and thus to increase the availability of the cutting tool. Replacing the tool only at failure may leave undesired effects on the product’s quality characteristics, namely, the dimensions and the surface finish. This may lead to scrapping of the product. The poor tool condition may cause wastage of subsequent production resources and
the loss of customer’s goodwill.2 In general, the deter-mination of the optimal replacement time is considered an important economic factor in machining.3
The cutting tool cost represents around 25% of the total machining cost.4,5 The cutting tool failure repre-sents about 20% of machine down-time,6and replacing cutting tool earlier or later than necessary will cause either loss of valuable resources or scrapping of prod-ucts.7Moreover, the tool replacement policy is one of the important aspects of tool management.8For these reasons, finding the time at which the tool should be
1E´cole Polytechnique de Montre´al, Montreal, QC, Canada 2Institute of Aerospace Research, NRC, Montreal, QC, Canada 3University of Ontario Institute of Technology, Oshawa, ON, Canada
Corresponding author:
Yasser Shaban, Department of Mathematics and Industrial Engineering, E´cole Polytechnique de Montre´al, Montreal, QC H3C3A7, Canada. Email: [email protected]
replaced is fundamental. Much research tried to improve tool life in several ways. For example, Klim et al.3proposed a method to improve cutting tool life in machining using the effect of feed variation on tool wear and tool life. By changing feed rate, the reliability function is changed, and thus the tool life is changed. The Weibull distribution was used to fit the data. The experiment was conducted under constant cutting
speed. Balazinski and Mpako9 proposed an
improve-ment of tool life through using two discrete feed rates. The method depends on varying the feed rate through-out the cutting process. By varying the feed, the tool-chip contact area increases, the tool wear rate decreases, and consequently leads to improvement of the cutting tool life. The experiment was conducted under constant
cutting speed. Lin and Shyu10 concluded that using
variable feed machining, and constant cutting speed, when drilling stainless steel is a significant method for improving the cutting tool life.
Other researches tried to find the optimal replace-ment strategy by using proportional hazards models (PHMs) for modeling tool life, then using another tech-nique to find optimal strategy. For example, Mazzuchi and Soyer11used a PHM to assess machine tool relia-bility. Fully Bayesian analysis is used to find optimal machining conditions. Liu and Makis12 derived a for-mula to calculate the cutting tool reliability under vari-able cutting conditions. They used PHMs while considering the machining conditions as covariates. Liu
et al.13 extended the work by developed algorithm
based on stochastic dynamic programming for finding the optimal tool replacement times in a flexible
manu-facturing system. Ding and He14 used a PHM by
con-sidering vibration signals as a time-dependent
covariate. The author suggests that vibration signals are good indicators to tool wear. Reliability analysis based on feature extraction from tool vibration signals is introduced. They found remarkable relationship between the tool condition monitoring information and the life distribution of tool wear by using PHMs. Other research used classical Weibull distribution to fit tool life distribution. For example, Vagnorius et al.15 used the Weibull distribution to fit tool life distribution. The optimal replacement time for metal cutting is deter-mined from a total time on test (TTT) plot.
Some researchers tried to improve the cutting tool life by changing feed rates while the cutting speed is constant;3,9,10others consider the PHM as good model for tool life representation.7,11,16In most of these mod-els, it was assumed that the machining conditions have significant effect over the entire tool life, but finding tool replacement models is still unavailable. The objec-tive of this article is to find tool replacement optimiza-tion models which can be used in order to minimize the cost or to maximize the availability during turning TiMMCs under variable conditions. The PHM is used to model in order to find these models. The Cutting speed (v) and the feed rate (f) are treated as the models’ covariates. In section ‘‘Model description of a tool
operating in varying conditions,’’ a brief description of the PHM of a tool operating in variable conditions is introduced. In section ‘‘Optimal replacement policy,’’ the optimal replacement policy for minimizing the cost and maximizing the availability is described. In section ‘‘Description of the experiment,’’ the experimental pro-cedure which was carried out in order to collect data that are used for constructing the model is presented. The model developed and the final results are presented in section ‘‘Development of the model and results.’’ Practical use and sensitivity analysis are given in section ‘‘Practical use and sensitivity analysis.’’ Concluding remarks are given in section ‘‘Conclusion.’’
Model description of a tool operating in
varying conditions
In 1907, Taylor17 developed the classical relationship between tool life (T) and cutting speed (v). The Taylor tool life equation is v Tn= K, where K and n are experi-mental constants which depend on the machining con-ditions and the material of cutting tool and the part. The Taylor’s equation shows that the tool life is inver-sely proportional to cutting speed. Taylor’s extended equation including machining conditions, namely, the cutting speed v and the feed f, is given in Mazzuchi and Soyer.11This equation has the following form
T= C
vxfy
ð Þ ð1Þ
where C, x, and y are positive constants. Taylor’s extended equation considers only the machining para-meters but fails to consider the aging and the progres-sive wear of the tool’s effect.11 In order to take into consideration the tool’s age, the tool life T is considered a random variable. Due to the flexibility of the Weibull distribution, it is extensively used in modeling the tool life. The Weibull failure rate for a tool in constant operating conditions, that is, the speed and feed, is given as follows h(t) =b h t h b1 ð2Þ where b is the shape parameter and h is scale para-meter. In PHM, the failure rate of the cutting tool is not only dependent on the age of the tool but is also affected by covariates which describe the machining conditions.11Based on equation (2), the PHM consists of the failure rate as the product of a baseline failure rate h0(t), which is dependent only on the age of the
tool, and an exponential expression which is the linear sum of giZi, where Z represents the covariates of the machining conditions. The failure hazard rate at time t is expressed as in equation (3) h t, Z; b, h, gð Þ =b h t h b1 exp X m 1 giZi ( ) ð3Þ Shaban et al. 925
Using the Weibull model as a baseline function in modeling the tool failure was considered by Tail et al.7,
Mazzuchi and Soyer,11 and Makis16 This model is
sometimes called the Weibull parametric regression model. The covariates are the cutting speed (v) and the feed rate (f). The model is given in equation (4), where
m= 2 h t, Z; b, h, gð 1, g2Þ =b h t h b1 eg1v+ g2f ð4Þ In this article, we consider two states: the normal and the failure states. This latter is defined by the tool wear reaching a predefined level VBBmax= 0:2 mm.The
survival function can thus be given as in equation (5) R t; Zð Þ = P T . tjZð Þ = exp H(t, Z)f g = exp ðt 0 h t, Zð Þdt 8 < : 9 = ; = exp t h b eg1v+ g2f ( ) ð5Þ where H(t, Z) is the cumulative hazard function. The
survival function R(t; Z) and its derivative
_
R(t; Z) = h(t, Z)R(t; Z) are used to estimate the para-meters (b, h, g1, g2) by using maximum likelihood
(ML) function.18
Optimal replacement policy
The classical age replacement strategy recommends replacement of the cutting tool at failure, that is, when the tool wear threshold is reached, or when it reaches a certain age which minimizes the cost per unit time. In the classical strategy, the effects of the covariates are not taken into account. In this article, the effects of the cutting speed (v) and the feed rate (f) are taken into con-sideration. The failure hazard rate of the cutting tool is a non-decreasing monotonic function, so the control-limit is used to find the minimum expected cost per unit time.19,20 The control-limit is a control-limit value (d . 0). The optimal stopping rule is given in equation (6). The stopping rule is often used in condition-based maintenance (CBM) as an alarm when uncontrollable covariates reach predefined states. In this article, it is used as follows
Td= inf t50 : Kh(t, Z)5df g ð6Þ
where Td is the preventive replacement time and K is the difference between the failure replacement cost
C+ K and the preventive replacement cost C.
According to the theory of renewal reward processes, the expected cost per unit time can be expressed as
fðTdÞ = C P Tð d\TÞ + (C + K) P(Td5T) W(d) =C+ K P(Td5T) W(d) ð7Þ d= f(T
d) is the optimal cost at which the f(Td) is
minimum and T
d is the optimal time to replace.
P(Td5T) is the probability of failure replacement, P(Td\T) is the probability of preventive replacement, and W(d) = E( minfTd, Tg) is the expected replacement time. Optimal level dcan be found by using the
fixed-point iteration procedure18,20 or by using
Semi-Markovian Covariate Process.21
Similarly, we represent the availability function as in equation (8) A(Td) = uptime uptime + downtime = W(d) W(d) + TpP(Td\T) + (Tp+ K) P(Td5T) ð8Þ When A(Td) is the availability. The optimal availability is achieved at Td the optimal time to replacement, Tp is the time required to perform the preventive replace-ment, and Tf= (Tp+ K) is the time required to per-form failure replacement. We note that in equation (8),
Kis the difference between Tfand Tp, while in equation (6), it is the difference between the failure replacement cost and the preventive replacement cost.
The objective is to find d. The replacement function is derived when dis obtained and the machining condi-tions, namely, the cutting speed (v) and the feed rate (f), are known. The replacement function is derived from equation (6) as follows Kh t, Zð Þ5d ð9Þ b h t h b1 eg1v+ g2f5d K ð10Þ eg1v+ g2f5d hbt(b1) Kb ð11Þ g1v+ g2f5ln dhb Kb b 1ð Þ ln t ð12Þ Zc5g(t) ð13Þ
g(t) was defined as a warning function by Banjevic et al.18 The function g(t) = ln (dhb=Kb) (b 1) ln t
can be considered as ‘‘replacement’’ function. By calcu-lating an ‘‘overall’’ covariate value Zc, the optimal time to replacement T
dis obtained.
Description of the experiment
Equipment: A 6-axis Boehringer NG 200, computer numerical control (CNC) turning center is used in order to conduct experiments, as shown in Figure 1.
Tool material: TiSiN-TiAlN nano-laminate physical vapor deposition (PVD) coated grades (Seco TH1000 coated carbide grades) is used. Workpiece material: A cylindrical bar of Ti-6Al-4V alloy matrix reinforced with 10%–12% volume fraction of TiC ceramic parti-cles is used. Experimental details: The experiments were conducted using full factorial designs with two factors,
two levels (v = 40, 80 m=min and f= 0:15,
0:35 mm=rev), and using one center point
(v = 60 m=min and f = 0:25 mm=rev). Full factorial designs are the most conservative of all design types because we try all combinations of the factor settings. Table 1 shows the design of the experiment in a coded form. Table 2 shows all combination of cutting condi-tions. There are five runs which were done randomly. Each run was replicated at least 5 times.
The cutting tool fails when the tool becomes dull and no longer operates within acceptable quality.5The com-mon way of quantifying the tool time to failure (TTF) is to put a limit on the maximum acceptable flank wear,
VBBmax. For each tool, sequential inspections were
conducted in order to measure the wear. The wear is monitored at discrete points of time through inspec-tions. The wear is measured after each inspection by using an Olympus SZ-X12 microscope. The procedure
continues until the tool wear threshold
(VBBmax= 0:2 mm) is reached. The procedure is
repli-cated for 28 tools.
Figure 2 shows the wear interpolation procedure in order to calculate the TTF; the wear evolution between two measurements (VBi, VBi+ 1) is assumed to be
lin-ear. TTF is calculated when tool wear threshold
(VBBmax= 0:2 mm) is reached. For example, from
Table 3, by interpolating between the 14th inspection at (ti= 1530 s) and the 15th inspection at (ti+ 1= 1650 s),
and by using equation (14), the TTF is found to be 1623.3 s. This interpolation is repeated for 28 tools. The results for the 28 tools are given in Table 4
d
Dt=
0:2 VBi
DVB , TTF= e + ti ð14Þ
Development of the model and results
The PHM parameters are estimated using EXAKT software.18 The resulting hazard function is given as follows in equation (15) h t, Zð Þ = b h t h b1 eg1v+ g2f = 3:71 23760 t 23760 2:71 e0:195v + 10:86f ð15Þ
The covariate parameters g1= 0:195 and g2= 10:86 are the multipliers for cutting speed (v) and the feed rate (f), respectively, in the hazard function. A small value
Figure 1. The experimental setup.
Table 1. The coded design of experiment.
Run Factor Cutting speed (m/min) Feed rate (mm/rev) Depth of cut (mm) 1 21 21 1 2 1 21 1 3 21 1 1 4 1 1 1 5 0 0 0
Table 2. The design of experiment.
Run Factor Cutting speed (m/min) Feed rate (mm/rev) Depth of cut (mm) 4 80 0.35 0.2 3 40 0.35 0.2 1 40 0.15 0.2 2 80 0.15 0.2 5 60 0.25 0.2
Figure 2. Wear interpolating.
for g1 parameter does not mean that cutting speed (v) has a small effect on the hazard function because the covariate parameter is multiplied by the covariate value which can be large.22 In order to distinguish between statistically significant and non-significant covariates, a formal statistical test is needed. In Table 5, statistical Wald test shows in column 5 that the cutting speed is more significant than feed rate.
In order to know how the cutting speed and the feed rate affect the hazard rate, a simple normalization pro-cedure is done. Since the cutting speed and the feed are
in the range (40, 80) and (0:15, 0:35), respectively, the normalization of the ‘‘overall’’ covariate will be as follows
Zc= bo+ b1x1+ b2x2= 14:415 + 3:9 x1+ x2 ð16Þ where x1= ((v 60)=20), x2= ((f 0:25)=0:1), and
x1, x22 ½1, 1.
bo, b1, and b2 are called regression coefficients.
23 In our model, it is obvious that the effect of cutting speed on cutting tool life is approximately four times more than the effect of feed rate.
In order to validate the model, Kolmogorov– Smirnov test (K-S test) and logarithmic reliability func-tion analysis are done. K-S test evaluates the model fit. The test checks the null hypothesis that the H(t, Z) in equation (5) is distributed exponentially (equation (13)). The summary of goodness of fit test is automati-cally produced in EXAKT as in Table 6.The test shows that the PHM offers a good modeling for the data.
Figure 3 shows the analysis of the logarithmic relia-bility function (log minus log plot).24 From equation (5), the linear equation for each run will be as follows
ln½ln R t; Zð ð ÞÞ = b ln tð Þ b ln hð Þ + g1v+ g2f ð17Þ The logarithmic reliability function in equation (17) is linear in ln(t), and for each run, corresponding func-tions are parallel.7It is concluded, now, that the PHM’s assumption is satisfied and the reliability functions of
Table 3. The experimental results showing the wear of tool 1–1.
Inspection no. Time (s) VB (mm)
1 0 0 2 120 0.0525 3 240 0.06 4 360 0.065 5 480 0.0725 6 600 0.0875 7 720 0.1075 8 840 0.1125 9 960 0.12 10 1050 0.125 11 1170 0.135 12 1290 0.165 13 1410 0.175 14 1530 0.1825 15 1650 0.205
Table 4. Time to failure (TTF) for the 28 tools.
Tool ID run-replication Time to failure (s) Speed (v), m/min Feed (f ), mm/rev Tool ID run-replication Time to failure (s) Speed (v), m/min Feed (f ), mm/rev 1–1 1623.3 40 0.15 3–5 1230 40 0.35 1–2 2087 40 0.15 3–6 1006 40 0.35 1–3 1770 40 0.15 4–1 121.4 80 0.35 1–4 1524 40 0.15 4–2 87.5 80 0.35 1–5 1560 40 0.15 4–3 135 80 0.35 2–1 295 80 0.15 4–4 135 80 0.35 2–2 267.5 80 0.15 4–5 121.7 80 0.35 2–3 281.2 80 0.15 4–6 102.5 80 0.35 2–4 225.3 80 0.15 5–1 233.5 60 0.25 2–5 252.7 80 0.15 5–2 192 60 0.25 3–1 1240 40 0.35 5–3 265 60 0.25 3–2 1002 40 0.35 5–4 190 60 0.25 3–3 1320 40 0.35 5–5 160 60 0.25 3–4 1263.3 40 0.35 5–6 185 60 0.25
Table 5. Summary of estimated parameters (based on ML method).
Parameter Estimate Significance Standard error Wald
Scale (h) 23,760 – 6174 –
Shape (b) 3.71 Y 0.6077 19.88
n 0.1951 Y 0.03356 33.78
the cutting tool in the range of the cutting speed and the feed rate are presented.
Based on equation (15), the failure rates are plotted for each run in Figure 4. The effect of machining condi-tions on the failure risk is clear when we compare between different runs. For example, by comparing between run 1 and run 2 which have the same feeds rates but different speeds, and also by comparing between run 2 and run 4 which have the same speeds but different feed rates, obviously, the effect of cutting speed is much higher than the effect of feed rate.
After determining the PHM, the optimal replace-ment policy-cost analysis is performed. The optimal replacement function is calculated with a cost ratio r = 2 (preventive replacement cost is estimated to be $100, and the failure replacement cost is $200, thus K is equal to $100); r is the ratio of the failure replacement cost to preventive replacement cost, and it is calculated considering the tool and material cost; r is always more than 1 to make sense to maintain the tool preventively. As shown in Figure 5, the optimal time to replacement
T
d can be calculated. The function
g(t) = ln (dhb=Kb) (b 1) ln t = 29:429 2:71 ln t
is the replacement function, applied to an ‘‘overall’’ covariate value Zc= 0:195 v + 10:86 f.
Similarly, we find the optimal replacement function that maximizes the availability. The optimal time to replacement Tdis then calculated. The time required to
perform preventive replacement Tp= 160 s, and the
time required to perform failure replacement
Tf= 540 s. As shown in Figure 6, the function
g(t) = ln (dhb=Kb) b 1ð Þ ln t = 31:22 2:71 ln t is
the replacement function, applied to an ‘‘overall’’ cov-ariate value Zc= 0:195 v + 10:86 f.
In practice, finding optimal replacement policy is generalized. Figure 7 shows the sequence of finding the optimal replacement T
d in both cases of cost analysis or availability analysis. For example, in cost analysis, the procedure is as follows
1. Extract the event (tool failure) by sequential
inspections for any machining process.
2. Collect the experimental data in order to build the model by estimating the parameters of the PHM.
3. Check the goodness of fit using, for example,
Kolmogorov–Smirnov test. 4. Find d= f(T
d) which is the optimal cost where f(Td) is minimum, and then find the replacement function, g(t) = ln (dhb=Kb) b 1ð Þ ln t for
known costs C and C + K. 5. Calculate T
d for current machining conditions v
and f by defining the composite covariate
Zc= 0:195 v + 10:86 f and using the replacement function.
Practical use and sensitivity analysis
The replacement function is used for a single cutting tool in multitasked machining process under variable
Table 6. Summary of goodness of fit test results.
Test Observed value P-value PHM fits data
Kolmogorov–Smirnov 0.2266 0.0965714 Not rejected
-30 -20 -10 0 10 1.6 3.6 5.6 7.6 ln[ -ln(R(t;Z))] ln(t) Run 1 Run 2 Run 3 Run 4 Run 5
Figure 3. Logarithmic reliability function plot for each run.
Figure 4. Hazard rate curves for each run.
0 5 10 15 20 25 0 500 1000 1500 2000
The composite covariates
Working age (sec)
Figure 5. Optimal replacement function-cost analysis.
machining conditions. For example, the user may use the tool for machining a part with machining condi-tions v = 50 m=min and f = 0:20 mm=rev for 200 s, then he or she may want to use the same tool for a sec-ond machining process with machining csec-onditions
v= 40 m=min and f = 0:15 mm=rev. The question is
‘‘Can he/she use this tool for the second machining pro-cess and for how long he/she can use this tool before
replacing it with a new one in order to get the cost optimality?’’ Figure 8 answers this question. The first machining process starts at point 1 while Zc= 11:92 and continues horizontally until point 2 (Td= 200 s). Since Point 2 is below the replacement function curve, the user can use the tool for the second machining pro-cess which will start at point 3 while Zc= 9:43 and can go horizontally until it touches the replacement func-tion curve (point 4), which gives the optimal time to replacement Td= 1610 s. The optimal remaining time for the second machining process is Td Td= 1410 s. Obviously, that example shows how the user can follow the status of the cutting tool by knowing its cutting speed v, feed f, and working age.
Sensitivity analysis is performed on the cost ratio (r). Figure 9 shows the cost ratio sensitivity when r = 2 to r = 5. Obviously, the optimal time to replacement is decreasing when the cost ratio (r) is increasing. This is very logical because as the difference between the fail-ure replacement cost and the preventive replacement cost gets higher, the more frequent preventive replace-ment should be done, thus the new optimal time to
0 5 10 15 20 25 0 500 1000 1500 2000
The composite covariates
Working age (sec)
Figure 6. Optimal replacement function-availability analysis.
replacement will be less than the original one, in order to minimize the cost per unit time.
Conclusion
In this article, we have introduced two new contribu-tions to the research on tool replacement, which are two optimality models for cost minimization and
availability maximization, and we applied it to a new generation of composites, namely, the TiMMCs. Experimentally, data were collected during turning TiMMCs under variable machining conditions. The collected data were used to construct the PHM. The PHM offered a statistically good model for the prob-lem. An optimal replacement function was obtained and built into a simple chart. While changing the machining conditions, we showed how the user can find the optimal time to replacement that optimizes either the machining cost or the availability per unit time. In these cases, the machining cost per unit time and avail-ability were found to be equal to 0.13 $/s and 80.77%, respectively. If these models are not used, the run to failure cost and availability are found to be 0.20 $/s and 64.57%, respectively. This represents a saving of 35.7% in case of cost analysis and an increasing of 16.20% in the case of availability analysis.
Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publi-cation of this article.
Figure 8. Optimal replacement example-cost analysis.
0.00 5.00 10.00 15.00 20.00 25.00 30.00 0 500 1000 1500 2000 2500 The composite covariates
Working Ɵme (sec) r=2,K=100 r=3,K=200 r=4,K=300 r=5,K=400
Figure 9. The cost ratio sensitivity.
Funding
The author(s) disclosed receipt of the following finan-cial support for the research, authorship, and/or publi-cation of this article: This research is funded by Natural Science and Engineering Research Council of Canada (NSERC), and Natural and Technology Research Fund of Quebec (FQRNT).
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Appendix 1
Notation
A availability
C preventive replacement cost
C+ K failure replacement cost
d control-limit value
d optimal cost
f feed rate
g warning function
h(t) failure hazard rate at time tð Þ
h0(t) baseline failure rate
H cumulative hazard function
P probability
r cost ratio
R survival function
t cutting time
T cutting tool life
Td preventive replacement time
T
d optimal time to replacement
v cutting speed
VB tool wear
VBBmax tool wear threshold
W expected replacement time
Z covariates of the machining conditions
Zc overall covariate b shape parameter bo, b1, b2 regression coefficients h scale parameter
